Any normative theory that attaches weight to the impartial value of an action’s consequences needs an axiology, a criterion for comparing worlds in terms of overall betterness. Though many different axiologies are available for comparing finite worlds, it has proven very difficult to extend these axiologies in a satisfactory way to infinite worlds. This is an urgent problem for moral philosophy, since it seems increasingly likely that our world is in fact infinite. In this paper, I show that the problem is even more difficult than has so far been appreciated. Specifically, while there is a natural way of extending finite axiologies that delivers ideal results for agents who can only make a finite difference to an infinite world, this extension fails for agents whose choices make an infinite difference to the distribution of value in the world. If there are infinitely many value-bearing entities in our causal future, then we almost certainly find ourselves in this position, since our choices will be “identity-affecting” with respect to infinitely many of those future beings. There is a very natural anonymity principle that seemingly must be true if we are able to make any axiological comparisons at all between the outcomes of infinitely identity-affecting choices. But in combination with other very plausible assumptions, this anonymity principle results in axiological cycles. From this cyclicity problem, I draw out several simple impossibility results indicating that, if the population of the causal future is infinite, then we will have to pay a very high theoretical price to hang onto the conclusion that our actions to matter from an impartial perspective.